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#1 streek405  Icon User is offline

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Calc 3 HW - Differentials

Posted 13 November 2013 - 02:46 PM

Can someone please tell me what I'm doing wrong for this problem:

If R is the total resistance of two resistors connected in parallel with resistances R_1 and R_2, then R = \frac{R_1 R_2}{R_1 + R_2}. If R_1 is measured to be 15 ohms with a possible percentage error of 3 percent while R_2 is measured to be 35 ohms with a possible percentage error of 5 percent, use differentials to estimate the maximum percentage error in the calculation of R.

my work: R = 15*35/(15+35) = 10.5
soooo... 10.5 + 15(.03) + 35(.05) = 12.7
but my answer is wrong...why?

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Replies To: Calc 3 HW - Differentials

#2 jjl  Icon User is offline

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Re: Calc 3 HW - Differentials

Posted 16 November 2013 - 06:10 PM

Plug in the values into the equation that produce the biggest difference from the solution with zero percent error.
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#3 x68zeppelin80x  Icon User is offline

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Re: Calc 3 HW - Differentials

Posted 29 November 2013 - 12:18 PM

I have provided a full explanation of the process for obtaining the percentage in comment below. There are 2 function: A verbose function that explains every step, and a minified version, which is essentially the same thing but optimized.

/**
 *  Explanation:
 *
 *  R1_Resistance        16        // Given
 *  R2_Resistance        34        // Given
 *  R1_Possible Error     0.0300   // Given
 *  R2_Possible Error     0.0500   // Given
 *  R1_Delta              0.4500   // R1_Resistance * R1 Possible Error
 *  R2_Delta              1.7500   // R1 Resistance * R1 Possible Error
 *  R1_Coefficient        0.4900   // R2_Resistance^2 / (R1_Resistance + R2_Resistance)^2
 *  R2_Coefficient        0.0900   // R1_Resistance^2 / (R1_Resistance + R2_Resistance)^2
 *  R1_Error              0.2205   // R1_Delta * R1_Coefficient
 *  R2_Error              0.1575   // R2_Delta * R2_Coefficient
 *  Delta_R               0.3780   // R1_Error + R2_Error
 *  Total_Resistance     10.5000   // (R1_Resistance * R2_Resistance) / (R1_Resistance + R2_Resistance)
 *  Maximum_Error         0.0360   // Delta_R / Total_Resistance
 *  Answer               03.6000%  // Ta-da!
 */
public class Resistance {
	public static void main(String[] args) throws java.lang.Exception {
		double r1 = 15, r2 = 35, e1 = 0.03, e2 = 0.05;

		displayResult("Long Method", compute1(r1, e1, r2, e2));
		displayResult("Short Method", compute2(r1, e1, r2, e2));
		
		// Output:
		// -------
		// Long Method  : 3.6000%
		// Short Method : 3.6000%
	}

	public static double compute1(double r1, double e1, double r2, double e2) {
		double dr1 = r1 * e1, dr2 = r2 * e2;
		double cd = Math.pow(r1 + r2, 2);
		double c1 = Math.pow(r2, 2) / cd, c2 = Math.pow(r1, 2) / cd;
		double r1e = dr1 * c1, r2e = dr2 * c2;
		double deltaR = r1e + r2e;
		double totR = (r1 * r2) / (r1 + r2);
		double maxErr = deltaR / totR;

		return maxErr;
	}

	public static double compute2(double r1, double e1, double r2, double e2) {
		return ((r1 * e2) + (r2 * e1)) / (r1 + r2);
	}
	
	// Helper functions follow:
	
	public static void displayResult(String label, double result) {
		System.out.printf("%-13s: %s\n", label, toPercentage(result));
	}
	
	public static String toPercentage(double fractionalValue) {
		return String.format("%.4f%%", fractionalValue * 100);
	}
}

This post has been edited by x68zeppelin80x: 29 November 2013 - 02:36 PM

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